A General Theory of Free Agency

Guest contributor and long-time reader, Nick Lewellen, looks deeply at the real forces that drive free agency. A little economic theory goes a long way, so really take some time to read through this a couple of times. It will almost certainly add to your free agency enjoyment. – 42

Starting in the mid 50s, professional sports went to back to school. All of a sudden, professional academics became incredibly interested in professional sports leagues and their employees, fans, and cities. Economists seemed to have the greatest interest in professional sports followed by statisticians, mathematicians, and game theorists. Of course, a lot of you already know about this from the movie (or the book) Moneyball.

Moneyball did some good things for sports fans. It introduced fans to things like advanced statistics and econometrics, the application of mathematical techniques to measure and test economic models, to measure the productivity of individual players. Simply put, Moneyball took some of the insights of academia and brought them to sports fans, and it gave us a more educated fan base.

Unfortunately, movies and books haven’t covered everything that sports economists and game theorists have published about professional sports, most of which would be very useful to the casual sports fan. This article is going to outline some of the basic insights of game theory and sports by providing a very general theory or model of free agency. This may not sound very exciting (it is the offseason), but economics and game theory can provide an analytical framework for strategic interactions or games that allow for a more nuanced and intelligent understanding of how front offices and players’ agents think.

I want this article to serve as a basic introduction and to provide you guys with some analytical tools so you can think more constructively about the free agency process. Hopefully, this will even lead a few of you to read more about economics and game theory in sports. One last note before I begin, most articles on sports and game theory are sure to include a lot of mathematical modeling, even if it is just a general case. I’m going to stay away from that entirely to prevent anyone from becoming too intimidated by the subject mater. I also want to stay away from using examples due to space constraints.

How does Free Agency Change Things?

I don’t think it is necessary for me to define what free agency is. I think every dedicated Bourbon Street Shots reader understands the free agency period and how it functions. I do want to spend a bit of time thinking about why free agency exists and how it might affect the allocation of players, and therefore wins, across teams. First, why do we have free agency? Really, it is all about giving the players more power to determine where they play and for how much. I’m not making a value judgment about this. I’m simply saying it’s a fact. For players, this is a pretty good deal. They are free to go to the highest bidder (or not) and don’t have to spend a career on a losing team or in a city they hate (or they can). Additionally, free agency does generally raise player salaries through bidding wars, which also makes it a great institution for players (Rottenberg, 1956; Depeken, 2011).

That last sentence answers some of the second part of our question. Free agency leads to higher salaries for players, which also decreases the profit earned by owners. As fans, we probably won’t shed many tears over an owner losing the little bit of profit they might have made. The one thing that should concern is how free agency changes the competitive balance in a professional sports league, especially for small market teams. Economic theory tells us that in a general case the competitive balance of professional sports leagues is not affected by free agency, because in any case the team that values the player the most will end up with him, either through a trade or free agency. According to most sports economics research on the NBA, this is probably true. Now, I don’t want to say I disagree with this research, but this research is all focused on long-term averages and seem to ignore the distinction between the hard and soft cap. All in all, I would say that it probably has had a minimal to moderate bias towards major media market teams but not a strong bias.

Free Agency as a Game

In economics or game theory, a game is any situation where two or more individuals (called players) make strategic decisions and moves to increase or secure their preferred outcomes. Using this definition, tic-tac-toe is a game, albeit a determined one (that means we always know what the best move in tic-tac-toe is at any point in the game). Basketball is a game, though it is obviously much more complex than tic-tac-toe. The free agency period is also a game. It is also much more complex than tic-tac-toe, and, like basketball, it contains layers of strategic interaction. Thinking of free agency as a game is important, because we can now take a rational cost benefit approach to free agency that will lead us to the most optimal decisions for any franchise. Before we get that point, I want to construct this game a little more clearly.

Who Plays the Game?

There are obviously two competing groups of individuals, the team and the player. However, teams are also competing with each other to land free agents, and players often compete with other players to land a contract. Of course, teams compete with other teams and players compete with other players for contracts, but for the time being, we are only interested in the interaction between the free agent, his representation, and the team. Their interaction and bargaining is called a negotiation game (also, a bargaining game).

What Does Each Player Want?

The first thing to determine is what each player wants to achieve or what their preferences are. This is also where most game theoretic modeling of free agency breaks down. In game theoretic modeling, you want both players to compete for the same type of payoff so you can compare their results. Unfortunately, teams and players do not try to maximize the same things.

Players are primarily trying to maximize the value of their contract. That is to say they want the most money possible. Obviously, players consider other things like a team’s chances of winning, long-term security, and location. However, these are almost always second order preferences and aren’t nearly as important to the average player as the value of the contract. I’m sure some will still try to argue and say some players care about winning more than anything else. That maybe true for the odd play here or there, but it isn’t the average or general case. In fact, most players will only take a slight pay decrease for a longer contract or for the chance at championship.

Here is where the model breaks down for most sports economists. What do teams want to maximize? Well, to reach a determined mathematical model we would have to say their profits (and a few papers argue that, see Solow and Krautman, 2011). I imagine most of us would argue that this is a second order preference for teams. Obviously, teams and their owners want to win first off. Professional sports just aren’t a profitable business for owners. It is all about winning. But how do we graph or mathematically represent wins in terms of real dollars? It is a hard thing to do, so most economics papers just look at pro sports teams as profit maximizing institutions.

One way to get around this problem would be to think of wins costing some amount of money. Since the salary cap is the main constraint faced by GMs’ determining player salaries, I often think of the value of a win being determined by the cap. So if a team has 58 million dollars to spend and thinks it will take at least 58 wins to get the playoff spot they want*, then the team wants to allocate their resources so that the spend about a million dollars for each win. A player who adds about 10 wins to a team a year will be worth a contract that averages 10 million dollars per year. Again, this is just one way to think about the value or cost of a win.

How Do They Get What They Want?

Now that we know what each side wants, we need to think about how they are going to interact to increase that amount. The player’s side is a bit more obvious but less interesting for Pelicans fans. The player and his agent are going to engage in negotiation strategies to increase their value. They may signal directly to a team or indirectly through the media that they are negotiating with other teams to increase teams’ offers. The player’s representation may try to increase their standing with the fan base in hopes of promoting a backlash against the team if he were to leave. In any case, the player and his agent do what they can to get a bigger contract.

But what about the teams? The truth is NBA teams probably take multiple approaches to come to an informed decision about the draft and free agency. There are several methods a team could use to negotiate with players. Teams could be preemptive and offer more than players expect, or they could wait until the end of the period and pick up players who could be desperate for a contract. Again, there are a lot of different options. One thing is certain though; each team uses a lot of complex analytics and data analysis. Then, they go through the same negotiation game as players. Their process is just more involved, because teams are trying to maximize something other than profit. They are trying to maximize wins, as well.

How Do Teams Maximize Wins?**

First of all, there are a lot of different ways to measure how much a player contributes to a team in terms of wins. The most direct way would be using multiple advanced statistics, specifically something like win shares or wins produced. Wins produced is a slightly lesser known model created by renowned sports economist David J. Berri. He has a blog called Wages of Wins that explains this metric. Basically, these measures aim to represent how many wins players add to their teams. They certainly have their faults, but they are still a good introduction to the concept of what a player truly contributes. The important thing to recognize is that we need some type of metric or standard for measuring how much each individual player contributes to the team, so we can determine his value to the team.

Now, imagine, all the guys on a hypothetical roster ordered by their contribution to the team. Some of them have expiring contracts and offers from other teams, while others may have a contract with your team, but they are under producers. Free agency is a trade off problem. A GM has to weigh the cost and benefits of losing or adding a player. For example, is it worth losing one of your players to another team if you have to increase his contract to keep him? Of course, this depends on how much the player contributes to your team and if his new contract will cost more than he contributes. The same type of logic works for adding a new player. A GM’s job is to make decisions about these types of problems given certain gray areas.

In the offseason, a GM also has to think about how each loss, addition, or trade will affect not just that one position or player, but also how it will affect the team’s rotation, team’s development, and its overall skill set. This entire section may have sounded simple and patronizing, and in a way this is the simplest way to introduce these concepts. However, I honestly don’t believe this type of logic and reasoning is common for NBA front offices or fans for that matter.

Let’s say your team has a serious weakness at small forward. He is your lowest contributor, and he only added three more wins to your team than if your back up had taken his minutes. Your team also doesn’t have a serious three point shooter or defender. Well, now we know a) which position we would like to upgrade and b) what skills we want to upgrade. When we go out and look for a player to meet those qualifications, remember that adding a new player will affect the rotation of your entire roster. How many minutes does the new guy normally get? Is it realistic to expect a career bench player to come in and contribute at the same level as a starter? What happens to your current starter? What happens to your bench?

These questions are important, but there is the one ultimate question for an NBA GM: How much will I pay this new player per expected win? That is the million-dollar question without a doubt. If he adds two more wins to your team, how much more are you willing to pay for that? The answer isn’t always as clear as it may appear to some. It can get messy, but that is the fun of it as a fan. That is why we have podcasts, articles, and games. There is always some gray area. Just remember this type of thinking next time you argue with a friend over who the Pelicans should draft or pick up in free agency.

Last Word

When I first got the idea for this article, I was going to include so much more, but it is a bit difficult writing about this stuff. I don’t want to skip anything or confuse anyone, and I really don’t want it to feel like a lecture. I have talked to Michael and others about making this a short term continuing series (i.e. something on game theory and sports economics during the offseason). I would just write on recent topics in the academic literature or respond to your questions. Before I continue, I’d like to hear from you guys. Leave something in the comments or shoot me a tweet @nicklewellen. If you have a question or idea that relates to sports economics and game theory, send it to me. I’d be happy to write on it.

*: I picked these numbers for the easy math.
**: A great paper showing a general mathematical model for optimizing NBA team behavior can be found here.

This is a fantastic article, more please. I want to point out one thing though:
“Obviously, teams and their owners want to win first off. Professional sports just aren’t a profitable business for owners. It is all about winning.”
I don’t think this is true in all cases. We’ve seen plenty of examples of owners having ulterior motives, most commonly cost cutting, while owning a team. That may not be true when they first buy the team, but when financial hardship strikes (I’m looking at you, Maloofs) winning often takes a backseat.

bigindian I intentionally stated that idea quite strongly, because I think it is close to an axiom in the real world.
Still, you are right that winning can take a back seat to cost cutting, but I believe these are only extreme cases. Even your example of the Maloofs is an extreme case, which I think you would agree with. In short, you could edit that statement in the article to begin with the words, “Except for extreme cases..”
In any case, I think your point has more to do with real world examples and complications. I was hoping to write a general or theoretical case, where know one buys a pro sports team for the profits. Still, in the real world there are occasional exceptions, however they are often in extreme financial situations. I guess I’m saying in the theoretical world I’m not sure you’re right, but in the real world you have pointed out an interesting caveat.

New City Awesome question. I honestly got a bit giddy when I saw it. For those who may not know, a first move advantage occurs in a game when a player would always chose to move first rather than second.
This may seem like a bit of a cop out, but I’m going with it depends. I have tired writing this a couple of times, and it keeps coming out super long. I’m just going to write a summary and you can ask something more specific if you like.
I think order for teams is more of a signal than anything. If you are the first team at a player’s door, you seem like you very serious about the player and you get the first word. You can also set the bar in terms of negotiation. If you are the last team you might want to come in and say, “all those things that those other teams told you is crap”. (i.e. the Dwightmare this year and the Lakers).
If we are talking the game against players hunting down contracts, I think we would see a first mover advantage for everyone but mega stars, who will get payed max deals in any case.
Again, I can give my thoughts in more detail if you’d like.

“Free agency leads to higher salaries for players, which also decreases the profit earned by owners.”
This seems logical, but it doesn’t really work in practise. The CBA specifies how much teams will spend on salaries through the salary caps, so while the individual free agent salaries can be inflated by this process, the TEAM salary should not be affected that much. Put it this way, if the hornets had to pay Davis say $10m next season instead of the rookie scale, would they automatically spend more next season? Probably not, they’d just spend less on incoming free agents instead.

kierantk I think what we have here is failure to communicate. The rookie wage scale and the free agency period are not exactly the same thing, so I ‘m not sure the example of Davis makes a lot of sense. But I think I might see the point you are trying to make.
I think you are suggesting individual salaries may inflate for certain players, but the salary cap keeps down the total team salary (yes?). I think this is obviously true. However, it is the salary cap that is making your point not the lack or presence of free agency. Basically, I think you are half right. You are right that the salary cap’s goal is to keep salaries in check. But imagine there was no salary cap in either case. Wouldn’t wages be higher when players are allowed to drive up their price through negotiation? I think of course.
A few published papers have seen a small inflation in player salaries in the NBA even with the cap. This is probably due to the NBA’s soft cap. In the MLB, there is an obvious inflation, and several famous papers in sports economics journals have shown that. The effect is not so great in the NBA or NFL.
Simply put, you are sorta right, but the soft cap confuses things. Also, this is a general case. It is a theory. It is not a description of reality. It is merely a theoretical framework. That may make it seem worthless, but concepts from “in a vacuum” theory can change how we think about the real world.
The paper by Depken does a good break down. Take it easy on me if this still isn’t clear. I’m quite tired, and I’m not sure I did a great job explaining this bit. If you want, I can point you in the direction of some published work that deals with this issue. Finally, I hope I don’t seem impolite or condescending, just trying to have a polite discussion. Thanks for your comment.
TL;DR He is right in some ways. Salaries are inflated but not as much as they would be without a cap. This is a theoretical model. Maybe, I will get in to the dirty bits of free agency in another piece.

Nicks65 kierantk
Thanks for your response. Yeah, I wasn’t too clear in what I was trying to say, but mostly that the owner’s bottom line isn’t impacted much since team salaries don’t rise, so much as the distribution of salaries under the cap is impacted through free agency.
I think the other interesting thing is that this inflation of the non-max vets (usually mid-level type players) leaves teams bidding for these players to have little space under the cap to pay for the lower salaried, but not quite as demanded players (probably solid bench players, but not starters). This could be where the smart GM’s can really make their impact, and it looks like Dell has been doing that which should bode well for building the team to contend